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Multiple Angles

Trigonometry is a very important subject without which the study of mathematics cannot be completed. The word "trigonometry" has been originated from two Greek-language words  "trigono" which meant "triangle" and "metron" meaning "measure".

This branch of mathematics deals with the relationships between sides and angles of right-angled triangles. There are 6 trigonometric ratios or functions that are commonly studied in trigonometry.

These ratios are sine (sin), cosecant (cosec or csc), cosine (cos), secant (sec), tangent (tan), cotangent (cot).

Initially, we deal with trigonometric functions of single angles, such as : sin$\theta$, cos$\theta$ etc. But later on, we come across with trigonometric functions of more than one angle which are termed as "multiple angles".

With a single angle function, the calculations are much easier than that of multiple angles, since it is easy to illustrate the behaviour of a single-angle function using a graph.

On the other hand, the m
ultiple-angle trigonometry is all about the expressions with angles like: 2x, 3y, 4$\theta$, nx, etc. The trigonometric functions have a property of being periodic i.e. repeating their patterns.

It is the fact that with the increase of multiplier, the number of solutions are also increased. 
There are various identities and formulas associated with multiple trigonometric functions. Let us go ahead and discuss about multiple-angle trigonometry in this article below.

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Double Angle Identities

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The trigonometric functions with double angles, i.e. 2x, 2A, 2$\theta$, such sin 2A, cos 2A, tan 2x etc are known as double angle trigonometric functions. The double-angles functions can be expressed in the form of single-angle functions. 

The identities related to double angle trigonometric functions are listed below:

1) sin2A = 2 sinA cosA

2) sin2A = $\frac{2\ tanA}{1+tan^{2}A}$

3) cos2A = cos$^{2}$A - sin$^{2}$A

4) cos2A = 1 - 2 sin$^{2}$A

5) cos2A = 2 cos$^{2}$A - 1

6) cos2A = $\frac{1-tan^{2}A}{1+tan^{2}A}$

7) tan2A = $\frac{2\ tanA}{1-tan^{2}A}$

Half Angle Identities

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In trigonometry, the single-angle functions can be written in the terms of half-angle functions. We obtain the half-angle trigonometric identities by simply replacing 2A by A and A by $\frac{A}{2}$.

The half-angle formulas are shown below:

1) $sinA$ = 2 $sin$ $\frac{A}{2}$ $cos$ $\frac{A}{2}$

2) $sinA$ = $\frac{2\ tan \frac{A}{2}}{1+tan^{2} \frac{A}{2}}$

3) $cosA$ = $cos^{2}$ $\frac{A}{2}$ - $sin^{2}$ $\frac{A}{2}$

4) $cosA$ = 1 - 2 $sin^{2}$ $\frac{A}{2}$

5) $cosA$ = 2 $cos^{2}$ $\frac{A}{2}$ - 1

6) $cosA$ = $\frac{1-tan^{2} \frac{A}{2}}{1+tan^{2} \frac{A}{2}}$

7) $tanA$ = $\frac{2\ tan \frac{A}{2}}{1-tan^{2} \frac{A}{2}}$

Triple Angle Identities

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The triple-angle trigonometric functions can also be expressed in the form of single-angle function. Let us see how to derive these formulas.

We are going to use the following sum-of-two-angles formulas:

sin(A + B) = sinA cosB + cosA sinB

cos(A + B) = cosA cosB - sinA sinB

sin 3A = sin (A + 2A)

= sinA cos 2A + cosA sin 2A

= sinA cos 2A + cosA 2 sinA cosA

= sinA cos 2A + 2 sinA cos$^{2}$A

= sinA (1 - 2 sin$^{2}$A) + 2 sinA (1 - sin$^{2}$A)

= sinA - 2 sin$^{3}$A + 2 sinA - 2 sin$^{3}$A

sin 3A = 3 sinA - 4 sin$^{3}$ A

Similarly,

cos 3A = cos (A + 2A)

= cosA cos 2A - sinA sin 2A

= cosA cos 2A - sinA 2 sinA cosA

= cosA cos 2A - 2 sin$^{2}$A cosA

= cosA (2 cos$^{2}$A-1) - 2 (1 - cos$^{2}$A) cosA

= 2cos$^{3}$A - cosA - 2 cosA + 2 cos$^{3}$A

cos 3A = 4 cos$^{3}$A - 3 cosA

The identity for triple angle for tangent can also be found by the above method. We have:

$tan3A$ = $\frac{3tanA - tan^{3}A}{1- 3tan^{2}A}$

Sine Multiple Angles

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The formula for multiple angles for sine is given below:
$sin\ nA = sinA\ (2 cosA)^{n-1} - \binom{n-2}{1} sinA (2 cosA)^{n-3} + \binom{n-3}{2} sinA (2 cosA)^{n-5} - ...$

By this formula, we get the following relations:

$sin4A = 4\ sinA\ cosA - 8 sin^{3}A\ cosA$

$sin5A = 5\ sinA - 20 sin^{3}A + 16 sin^{5}A$

$sin 6A = 6\ sinA\ cosA - 32 sin^{3}A cosA + 32 sin^{5}A cosA$

$sin7A = 7\ sinA - 56\ sin^{3} A + 112 sin^{5}A - 64 sin^{7}A$
and so on.

Cosine Multiple Angles

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The formula for multiple angles for cosine is given below:
$cos\ nA$ = $\frac{1}{2}$ $(2 cosA)^{n} - n(2 cosA)^{n-2}$ + $\frac{n}{4}$ $\binom{n-4}{1} (2 cosA)^{n-3}$ - $\frac{n}{6}$ $\binom{n-4}{2} (2 cosA)^{n-6} - ...$

By this formula, we get the following relations:

$cos4A = 8 cos^{4}A - 8 cos^{2}A + 1$

$cos5A = 16 cos^{5}A - 20 cos^{3}A + 5 cos A$

$cos 6A = 32 cos^{6}A - 48 cos^{4}A + 18 cos^{2}A -1$

$cos 7A = 64 cos^{7}A - 112 cos^{5}A + 56\ cos^{3} A - 7 cos A$

and so on.

Tangent Multiple Angles

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Tangent multiple angles can be deduced by dividing sine multiple angle formula by corresponding cosine multiple angle formula, i.e.

$tan nA$ = $\frac{sin nA}{cosnA}$

We get the following identities:

$tan 4A$ = $\frac{4 tanA - 4 tan^{3}A}{1-6tan^{2}A+tan^{4}A}$

$tan 5A$ = $\frac{5 tan A - 10 tan^{3}A + tan^{5}A}{1-10 tan^{2}A+5 tan^{4}A}$
and so on.
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